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Thermal recti?cation in carbon nanotube intramolecular junctions: Molecular dynamics calculations

Gang Wu?

Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542-76, Republic of Singapore

Baowen Li?

arXiv:0707.4241v1 [cond-mat.mes-hall] 28 Jul 2007

Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542-76, Republic of Singapore and NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Republic of Singapore (Dated: February 1, 2008) We study heat conduction in (n, 0)/(2n, 0) intramolecular junctions by using molecular dynamics method. It is found that the heat conduction is asymmetric, namely, heat transports preferably in one direction. This phenomenon is also called thermal recti?cation. The recti?cation is weakly dependent on the detailed structure of connection part, but is strongly dependent on the temperature gradient. We also study the e?ect of the tube radius and intramolecular junction length on the recti?cation. Our study shows that the tensile stress can increase recti?cation. The physical mechanism of the recti?cation is explained.

PACS numbers: 66.70.+f, 44.10.+i, 61.46.Fg, 65.80.+n

I.

INTRODUCTION

In past two decades, the study of heat conduction in low dimensional model systems has enriched our understanding on heat conduction from microscopic point of view1 . In turn, the study has also lead to some interesting inventions for heat control and management devices. For example, the heat conduction in nonlinear lattice models2,3,4 demonstrates recti?cation phenomenon, namely, heat ?ux can ?ow preferably in one direction. Furthermore, the negative di?erential thermal resistance is also found and based on which, a thermal transistor model has been constructed.6 Most recently, the two segment model of thermal recti?er proposed in Ref 2,3 has been experimentally realized by using gradual mass-loaded carbon and boron nitride nanotubes.7 These works have opened a new era for heat management and heat control in microscopic level. On the other hand, there have been increasing studies on heat conduction in real nano scale systems8 . For example, the thermal conduction of single walled carbon nanotubes (SWCNTs) has attracted both theoretical9,10,11,12,13,14,15,16,17,18 and experimental19,20,21,22,23,24,25 attentions. Almost all experiments and numerical simulations have payed their attention to the extremely high thermal conductivity of SWCNTs. The dependence of the thermal conductivity on the length or chirality of SWCNTs is also extensively studied theoretically. Therefore, one may asks, how can we make heat controlling devices, i.e., the thermal diode or the thermal transistor, from SWCNTs and their derivatives? In the practical applications, SWCNTs may have many kinds of impurities such as element impurities, isotopic impurities26 , topological impurities, etc. Among these numerous derivatives of SWCNTs, the

SWCNT intramolecular junctions (IMJs), which are formed by introducing the pentagon-heptagon rings in SWCNTs, have been expected to be an important structure in the applications. It is well-known that the electronic properties of the SWCNT IMJs have a close relationship with their geometrical or topological characteristics.27,28,29,30,31,32,33,34 It is reasonable that the thermal transport behavior of SWCNT IMJs can also be altered by their geometrical characteristics. Compared with the electronic properties, the thermal transport behavior may have not so close relationship with the detailed local geometrical arrangements because it is mainly in?uenced by the long wavelength phonons. But the non-equilibrium thermal transport behavior of IMJs can be a?ected by the high-frequency optical phonon modes,35 which can re?ect the information of the local defects. So it is interesting to investigate the nonequilibrium thermal transport of di?erent SWCNT IMJs. In this work, the thermal recti?cation in (n, 0)/(2n, 0) IMJs and its dependencies on tube radius, IMJ length, and external stress, e.g., tensile and torsional stress are studied. A complex structure of ‘peapod’ structure and IMJ is also studied to investigate the e?ect of periodical potential on the thermal recti?cation. The paper is organized as the follows. In Sec. II we introduce the basic structure of (n, 0)/(2n, 0) IMJ and our numerical method. In Sec. III, the main numerical results are discussed. In Sec IV, we show that the external stress can improve the recti?cation. In the same section we also show the results of (n, 0)/C60 @(2n, 0) structures, which can be regarded as a combination of peapod structures and (n, 0)/(2n, 0) IMJs. In Sec. V, we give some concluding remarks.

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II. MODEL AND METHODOLOGY

A typical (n, 0)/(2n, 0) IMJ structure is depicted in Fig. 1, in which the index n equals to 8. The structure contains two parts, namely, a segment of (n, 0) SWCNT and a segment of (2n, 0) SWCNT. For simplicity, the lengths of the two segments are almost equal in our calculations. The two segments are connected by m pairs of pentagon-heptagon defects. Because (n, 0) and (2n, 0) tubes have the common rotational symmetry C2n , thus when n can be divided exactly by m, we can adjust the defects to make the two segments connect straightly. Especially, if m = n, the pentagon-heptagon defects can be arranged in the connection part so that the rotational symmetry of the IMJ is Cn . On the other hand, it is necessary to de?ne a basic system length because the thermal conductivity depends on the system length.26 After setting the lengths of (n, 0) and (2n, 0) tubes to be the same, we can de?ne a basic total length L0 when both of the two segments contain 24 periods. All the structures are fully optimized before further molecular dynamic (MD) calculations, and then we can obtain L0 ≈ 20 nm.

The thermal ?ux is obtained from the thermostats, i.e., the total work from the thermostats can be regarded as the heat ?ux runs from thermostats to the system. If the work is positive, then the heat ?ux is also positive. In the scheme of the Nos?-Hoover thermostat,36 the equation of e motion for the particle i in heat bath is: ˙ pi = ?ξpi + fi , 1 ˙ ξ= Q pi · pi ? gkb T mi

,

(1)

where pi is the momentum and fi is the force applied on the atom. g is the number of degrees of freedom of the atoms in the thermostat. Q = gkb T τ 2 , where τ is the relaxation time. In our simulation, τ is kept as 4 ps. The heat bath acts on the particle with a force ?ξpi , thus the i ·p power of heat bath is ?ξ pmi i , which can also be regarded as the heat ?ux from heat bath, i.e., pi · pi . mi

Ji = ?ξ

(2)

FIG. 1: (Color online). A typical (n, 0)/(2n, 0) structure. Here, n = 8, and the number of pentagon-heptagon defects is m = 4. The regions marked as ‘I’ are ?xed in MD process. The regions marked as ‘II’ are put in the heat baths.

The total heat ?ux from the heat bath to the system can be obtained by J = i Ji , where the subscript i runs over all the particles in the thermostat. The ?nal j·s JL thermal conduction is κ · s = 2?T /L = 2?T , where j is the heat ?ux density, s the area of cross-section, and L the system length. In fact this de?nition is equivalent d to the usual de?nition: J = dt ri (t) εi (t), εi (t) is the

i

instant total energy of particle i, but Eq. (2) is much simpler for computational simulation.

III. NUMERICAL RESULTS AND DISCUSSIONS

In this kind of structure, the outmost one period of each heads (colored by blue and marked as region ‘I’ in Fig. 1) are ?xed in MD process. Then two periods of each ends (illustrated by orange color and marked as region ‘II’ in Fig. 1) are put in the heat baths, which are realized by the Nos?-Hoover thermostat.36 The tempere atures of the thermostats at left and right heads are TL and TR , respectively. For convenient, here we introduce two quantities: ?T = TL ?TR , and T = TL +TR . In this 2 2 work, T is always kept at 290 K. The C-C bonding interactions are described by the second-generation reactive empirical bond order (REBO) potential,37 which is the most recent version of the Terso?-Brenner type potential, combining advantages of the two sets of parameters in the earlier version.38 The velocity Verlet method is employed to integrate the equations of motion with the time step of 0.51 fs. The typical total MD process is 5×106 steps, which is about 2.55 ns, and the statistic averages of interesting quantities start from half of the MD process, i.e., 2.5×106 steps are used to relax the system to a stationary state. The instant temperature of atoms i is de?ned as 2 2 2 m Ti (t) = 3ki vx (t) + vy (t) + vz (t) , where v(t) is the b time-dependent velocity, mi the mass and kb the Boltzmann constant. This is a result of energy equipartition theorem.

First of all, we investigate the e?ect of defects in connection region. The defects in connection region determine the property of the interface between two tubes, and as a result, they will a?ect the heat transport property of the system. However, the question whether they can a?ect the recti?cation is still open. We study the (8, 0)/(16, 0) structures as examples. Generally speaking, they are many methods to connect an IMJ, thus we only consider two conditions here. The ?rst one is the coaxal straight IMJs. The possible number m of pentagon-heptagon defect pairs for a coaxal IMJ is 8, 4 or 2, but m = 2 makes the cross-section of the tube part highly deformed from circle, so we only consider m = 4 and 8. The second condition is the simplest IMJ, i.e., two tubes are connected with each other by using only one pair of heptagon-pentagon defects (m = 1).39 The top views and side view of the three structures (m = 8, 4 and 1) are plotted in Fig. 2. It can be found even in the simplest zigzag/zigzag IMJ (m = 1), there is still a small angle between the two segments. The total heat ?uxes J versus the normalized temperature di?erence ?T / T is plotted in Fig. 3a. Positive

3 for m = 4 is larger than that for m = 8. It is straightforward because the pentagon-heptagon defects are topological defects, they scatter the phonons and cause larger interfacial thermal resistance. As a result, more topological defects will reduce the thermal conductance and the heat ?ux. But the total heat ?ux for m = 1 is close to that for m = 4. This is due to the big di?erence of the connecting region of these IMJs. The length of the connecting region in IMJ with m = 1 is much longer than those in the IMJs with m = 4 or 8. The long connecting region can cause big thermal resistance, as a result, the thermal resistance in m = 1 IMJ is not much smaller than that in m = 4 IMJ. When the temperature di?erence is very small, the heat ?ux only shows weak asymmetry. The asymmetry becomes larger when temperature di?erence is larger. This characteristic is seen clearer in Fig. 3b, in which the thermal recti?cation |Jn /Jp | = ?Jn /Jp is drawn. In Fig. 3b, Jn means that the heat ?ux is of negative sign, vice versa, Jp means that the heat ?ux is of positive sign. It can be seen that, an increase in temperature gradient |?T | leads to an increase of thermal recti?cation. This is because the recti?cation is a result of non-equilibrium transport, which is mainly determined by optical phonons. Only when the temperature di?erence is large enough, the optical phonons can be excited and contribute to the heat conduction. More interestingly, when the temperature di?erence is large enough (for |?T | ≈ 0.8 T ), the recti?cation begins to decrease. But, the temperature di?erence is so large that this result may not be hold in more restrict calculations due to lose of local thermal equilibrium and the quantum effect in the lower temperature head. Another meaningful feature is that, although the absolute values of the heat ?uxes are completely di?erent, recti?cations are almost the same in IMJs with m = 4 and 8, whose connecting region is short. Namely, the recti?cation is weakly dependent on the detailed structure of the interface when the connecting region is short enough. And the recti?cation in m = 1 IMJ is largest among the three kinds IMJs. In order to understand above recti?cation phenomenon, we consider the power spectra of the atoms around the connecting part. The atoms to be studied are illustrated in the Fig. 4a. They are selected to be next to the defects along the axes. The corresponding phonon spectra are presented in the Fig. 4b. And the overlapping of the phonon spectra of the two atoms around the connecting parts have been emphasized by shadows. It is found that when the direction of the temperature gradient is exchanged, the area of the overlap region is also changed. More importantly, the overlap area is larger when ?T / T is negative. This result corresponds with the trend of the heat ?ux. So we can state that in this real system, the relationship between the overlap area and the absolute value of the heat ?ux is same as that in the one dimensional nonlinear lattice systems2,3,4 , in which it is found that matching/mismatching of the en-

FIG. 2: (Color online). The top views and side view of two (8, 0)/(16, 0) IMJs with di?erent connection methods. (a) With 8 pairs of pentagon-heptagon defects. (b) With 4 pairs of pentagon-heptagon defects. (c) With only one pair of pentagon-heptagon defects.

J means the heat ?ux ?ows from (n, 0) tube to (2n, 0) tube. We do not show the heat ?ux density here, because it is di?cult to de?ne the area of the cross-section of the whole IMJ.

FIG. 3: (Color online). The thermal transports of (8, 0)/(16, 0) SWCNT IMJs with di?erent numbers of pentagonheptagon defects in the connection region. (a) The heat ?ux versus temperature di?erence. (b) The thermal recti?cation versus temperature di?erence. m is the number of pentagonheptagon defects. Error bars are also plotted.

It can be found from Fig. 3a that the total heat ?ux

4 quency modes is weak because their group velocities are much smaller than those of low frequency modes. So the modes at 400?1000 cm?1 are more important than the modes at 1600?1800 cm?1 . The frequencies of the low frequency modes have been found to be inversely proportional to the tube radius.40 Thus, the low frequency modes, e.g., the radial breathing mode, have di?erent exciting temperatures in (16, 0) and (8, 0) tubes. On the other words, their participation in transport process occurs at di?erent temperatures. This fact further explains why these low frequency optical phonon modes are important for the recti?cation. Based on above discussions, now we can understand the following phenomena. Firstly, the recti?cation almost does not change in the m = 4 and 8 IMJs. Secondly, the recti?cation in m = 1 IMJ is largest in the three IMJs. As has been mentioned, the modes at 400?1000 cm?1 are most important for the thermal recti?cation. These modes have larger wave lengths than the high frequency modes. The connecting parts of the two IMJs with m = 4 and 8 are short, as a result, the defects have weak e?ect on the e?ective optical modes. In Ref. 41, it is found that the high frequency vibrational modes caused by topological defects are localized states. And the higher frequency localized modes show smaller spatial dimension, which means they can hardly entangle with the transport phonon modes. Thus in the short connecting IMJs (m = 4 and 8), the arrangement of the detailed defects has weak in?uence on the recti?cation. In contrast, the connecting region is much longer in the m = 1 IMJ, and the distance between the pentagon and heptagon rings is relatively long, which means the connecting region can a?ect those middle frequency phonon modes, which have longer wave lengths. Therefore, the recti?cation is largest in the case of m = 1 IMJ. However, the recti?cation in current structures is still small, which might limit the application of the IMJ as a thermal recti?er. So we will try to ?nd the factors which can help improve the recti?cation. Above all, we investigate the thermal conductance of the IMJs with the di?erent index n numerically. In order to make the results comparable with each other, the number m of pentagon-heptagon defect pairs are set to be n. Considering the computational consumption, only n = 7, 8 and 9 are considered in Fig. 5. According to Fig. 5a, the total heat ?ux increases when n increases, but this does not mean that the thermal conductivity for big n is also large because it should be scaled by the area of cross-section. In fact, it has been shown that the thinner SWCNTs have higher thermal conductivity in literatures.15,18,23 Another interesting feature is that, when index n increases, the asymmetric heat ?ux ratio does not always increase. In Fig. 5b, the largest recti?cation appears when n = 8. So this reminds us that the heat ?ux recti?cation is not only induced by the radius di?erence of two segments. When tube radius increases to a su?cient large value, i.e., n is large, the vibrational density

FIG. 4: (Color online). (a) The side view of the (8, 0)/(16, 0) IMJ (m = 8), where the atoms whose power spectra are recorded are emphasized by balls. (b) The corresponding phonon power spectra. The overlapping of the phonon spectra of the two particles around the connecting parts have been emphasized by shadows and bordered by dark lines.

ergy spectra around the interface is the underlying mechanism of the recti?cation. On the other hand, the most obvious change appears at 400?1000 cm?1 and 1600?1800 cm?1 . The m = 4 IMJ is also studied, and similar conclusion can be obtained. So the optical phonon modes are important for the recti?cation. In addition, in Ref. 35, the authors show that in carbon nanotubes with ?nite length where the longwavelength acoustic phonons behave ballistically, even optical phonons can play a major role in the non-Fourier heat conduction. The dispersion relations of the SWNT show that, in the intermediate range of the normalized wave vector 0.1 < k ? < 0.9, some of the phonon branches, especially the ones with relatively low frequency, have group velocity comparable to the acoustic branches. Furthermore, the thermal conductivity can be expressed as κ = lλq Cq (ωλ ) vλq , where λ is a set of quanq

tum numbers specifying a phonon state, l is the phonon mean free path, vλq is the magnitude of the phonon group velocity along the direction of the heat ?ow, and

exp( ω/kB T ) Cq (ω) = kB kBω is the thermal caT [exp( ω/kB T )?1]2 pacity of lattice wave with wave vector q and angular frequency ω. Thus, the contribution from the high fre2

5

FIG. 5: (Color online). The thermal transports of (n, 0)/(2n, 0) SWCNT IMJs, n = 7, 8 and 9. (a) The heat ?ux versus temperature di?erence. (b) The thermal recti?cation versus temperature di?erence. In this ?gure, m = n. Error bars are also plotted.

FIG. 6: (Color online). The thermal transports of (8, 0)/(16, 0) SWCNT IMJs with di?erent length. (a) The heat ?ux versus temperature di?erence. (b) The thermal recti?cation versus temperature di?erence. m = 4. L0 is de?ned in text. Error bars are also plotted.

of states (VDOS) of carbon nanotube approaches to that of the graphite. But when n is small, the VDOS of the SWNT deviates from that of the graphite due to the periodical boundary along the circumferential direction and the curvature e?ect, and the amplitude of the deviation is almost inversely proportion to n. As a result, when (n, 0) and (2n, 0) tubes connect with each other, the di?erence between the VDOSs of the two tubes is obvious if n is small, and vice versa, the di?erence is not so obvious if n is large. Recalling the fact that the recti?cation can be related with the mismatching between the VDOSs around the connecting region, we can conclude that the recti?cation of the IMJ is not obvious when n is large. Next, we investigate the length dependence of heat conduction. From Fig. 6a, it can be seen that the total heat ?ux decreases as the system length increases from JL L0 to 2L0 . However, the thermal conductivity, κ = 2s?T , increases with the increasing of the system length. This result agrees with the previous studies. Moreover, there still exists asymmetry in the heat ?ux, which is clearly illustrated in Fig. 6b. Where the recti?cation at di?erent |?T | is plotted for di?erent system lengths. Obviously, the increase of system length weakens recti?cation. It can be understood as follows. As the system length increases, long-wave phonons modes contribute more to the heat conduction. This kind of heat conduction is symmetric because long-wave phonon modes are hardly scat-

tered by the junction or any local defects. Moreover, the asymmetric thermal conductance is mainly controlled by the asymmetric interface, i.e., the connection part, which does not change when system length increases. As a result, although the absolute value does not change, the recti?cation appears to be smaller. Based on above study, we can conclude that changes in radial and axial direction have di?erent in?uence on recti?cation. The increase of structural asymmetry around the interface can increase the recti?cation. In the following section we will discuss other alternatives to improve the recti?cation.

IV. IMPROVEMENT

First, it is possible to increase the di?erence in force constants around the interface by applying external stress. Because the Young’s modulus has been proved to be almost the same in di?erent tubes using the Terso?Brenner potential,42 it can be expected that the (n, 0) tube and (2n, 0) tube will give di?erent response to the same external stress. Thus it is possible that one can use this property to fabricate devices. Here we consider two kinds of stresses, tensile and torsional stresses. It has been proven in the framework of tight-binding43 that these two stresses can change the electronic structure of

6 SWCNTs greatly. Here the (8, 0)/(16, 0) SWCNT IMJ (m = 4) is taken as an example. The original length is L0 . The stresses considered in this work are all small enough to ensure the system does not undergo any structural transformations44 or defects.45 For tensile stress, we elongate the IMJ along the axial direction by 3%. For torsional stress, one head of IMJ is rotated relative to the other by 30? around the axis. The results are shown in Fig. 7. the SWCNTs will shrink in the radial direction and elongate in the axial direction. This means their cylinder surface will change, so that the acoustic phonon modes will also be changed. But when SWCNTs are twisted, the cylinder surface almost does not change, so that the thermal conductance almost does not change. Comparing Fig. 7a and 3a, it can be found that the heat ?ux decreases in elongated structure. The reason is straightforward. As mentioned above, the thermal conductivity can be expressed as κ = lλq Cq (ωλ ) vλq ,

q

FIG. 7: (Color online). The thermal transports of (8, 0)/(16, 0) SWCNT IMJs under di?erent stresses. (a) The heat ?ux for di?erent stresses. (b) The thermal recti?cation for di?erent stresses. m = 4. Error bars are also plotted.

It can be seen that the torsional deformation can change neither the absolute value of heat ?ux nor the asymmetric behavior largely. However, the tensile stress can greatly change the heat transport. In the elongated structure, the heat ?ux becomes smaller than that of undeformed structure, but the recti?cation becomes larger. Above results are reasonable. First of all, the deformation of the thicker tube is smaller than that of thinner ones under same external stress. This means that when thick and thin tubes are connected with each other to form IMJ, they will appear di?erent axial deformation under the same stress, which will further increase the structural di?erence between the two tubes. Secondly, the heat transport in carbon nanotubes are mainly contributed by the four acoustic phonon modes. These phonon modes have weak dependence on the local structure, but they can be a?ected by the shape of the cylinder surface. When tensile stress is applied to the SWCNTs,

where Cq (ω) = kB For the low frequency ( ω ? kB T ) modes, which are most important for the thermal transport, Cq (ω) ≈ kB . Furthermore, lλq and vλq are the increasing functions of the force constants k, thus the thermal conductivity κ is also an increasing function of the force constants k. In fact, some numerical simulations show that κ ∝ k 2 in the weak coupled lattice models.2,46 When the system is elongated, the bonds along the axis are also elongated. Then according to the Abell-Terso? formalism of the Terso?-Brenner potential, the corresponding force constants will decrease. Finally, the thermal conductivity decreases in the elongated structure. To elongate an IMJ is not very di?cult for the stateof-the-art experiments, so we believe that tensile stress can be a possible method to improve the recti?cation. Next, we will try to modify the structure to change recti?cation. It is suggested2,3 that a suitable on-site potential can induce large recti?cation in two-segment model. So it is quite interesting that what will happen if some periodical potential is introduced into the SWCNT IMJs. On the other hand, a nanotube with su?ciently large diameter can be ?lled with spherical C60 fullerene molecules to build up a new hybrid structure referred to as a ‘peapod’,48,49 with spherical fullerenes representing peas and the carbon nanotube representing a pod. The thermal conductivity of the peapod structure has been investigated by using MD recently.50 So it is possible to combine these two structures, i.e., IMJs and peapod, into a new structure.

ω kB T

2

exp( ω/kB T ) . [exp( ω/kB T )?1]2

FIG. 8: (Color online). The (8, 0)/C60 @(16, 0) structure. The number of pentagon-heptagon defects is m = 4. The regions marked as ‘I’ are ?xed in MD process. The regions marked as ‘II’ are put in the heat baths.

The structure (see Fig. 8) contains two parts, i.e., a segment of (n, 0) SWCNT and a segment of (2n, 0) SWCNT with some C60 balls inserted into its center, which is typically called as SWCNT peapod structure. The lengths of these two segments are almost the same. They are connected by m pairs of pentagon-heptagon defects. Because the closest distances between the adjacent

7 C60 balls are almost same, 10.05 ?, we can consider that A they exert an external periodic potential on the outside SWCNT. Here, the interaction between two atoms on C60 balls and outside SWCNT is modeled by Lennard-Jones potential: σ r

6

V (r) = 4ε ?

+

σ r

12

,

(3)

with parameters, ε = 2.964 meV and σ = 3.407 ?. By A default, the total length of the structure is L0 , which permits 10 C60 balls in the (2n, 0) segment. In this complex structure, the two heads are closed to ensure the C60 balls not running out in the MD calculations. And several periods of two heads (presented by blue color and marked as region ‘I’ in Fig. 8) are ?xed in the MD simulation. Then two periods of each ends (emphasized by orange color and marked as region ‘II’ in Fig. 8) are put in the Nos?-Hoover thermostat.36 e In Fig. 9, the heat transports of (8, 0)/C60 @(16, 0) structure with di?erent lengths are shown.

this prevents the possible mass transport (via fullerenes) which in turn reduces the assistance transport e?ect of C60 balls. Furthermore, because of the coupling between the phonon modes (mainly the radial breathing modes) of carbon nanotube (CNT) and C60 balls, the conduction phonon will be scattered and the thermal conductivity of outside CNT will be reduced. On the other hand, we can notice that the length of error bars in Fig. 9b is much larger than that in Fig. 6b. This means that the existence of C60 balls increases the ?uctuations in the heat ?ux. In Ref. 50, it is found that the temperature ?uctuations of peapod structure are larger than that of empty nanotube. It is explained by the weak van der Waals forces between fullerenes and the outside CNT. Another result is that the existence of C60 balls can hardly change the asymmetric behavior of heat ?ux. This is somehow disappointing for us, because it is natural to expect that the periodical potential introduced by the C60 balls can enhance the thermal recti?cation according to the results based on lattice models. However, our numerical results are also reasonable because of following facts. The periodical potential introduced by the C60 balls is intrinsically van der Waals potential, so its strength is much weaker than the binding potential among the carbon atoms in SWCNT. The period of the periodical potential introduced by C60 balls is about 10 ?, which is too large compared with the C-C bond length A in SWCNT, thus only low frequency vibrational phonons are a?ected. As has been mentioned, the asymmetric heat ?ux is controlled by the optical phonon modes, so the change of low frequency vibrational modes can hardly change the asymmetric behavior. According to our result, if one wants to build a thermal recti?er by applying periodical external potential, van der Waals force is not a good choice.

V.

CONCLUSIONS

FIG. 9: (Color online). The thermal transports of (8, 0)/C60 @(16, 0) structures with di?erent length. (a) The heat ?ux versus temperature di?erence. (b) The thermal recti?cation versus temperature di?erence. m = 4. L0 is de?ned in text. Error bars are also plotted.

Comparing Fig. 9a with Fig. 6a, one can ?nd that the heat ?ux of peapod structure is smaller than that of the empty IMJ, which seems to be in contradict with the result in Ref. 50. But it should be pointed out that in our structure the peapod region is closed at the ends. In fact,

In this work, heat recti?cation in (n, 0)/(2n, 0) IMJs has been studied. It is found that the recti?cation depends weakly on the detailed structure of connection part, but depends strongly on the temperature gradient. The recti?cation increases as the temperature gradient increases. We have also studied the dependencies of the recti?cation on tube radius and IMJ length. We found that the maximum recti?cation appears in short (8, 0)/(16, 0) IMJ. Moreover, we have found that the tensile stress can increase the recti?cation, while torsional stress can not. It is due to the di?erent e?ects of the stresses on IMJ structure. It can be believed that the tensile stress will play an important role in building of devices with highly asymmetric transport behavior. The ‘peapod’ structure is combined with IMJ structure to investigate the e?ect of periodical potential on the

8 recti?cation. Our numerical result shows that the new structure can hardly change the recti?cation. The study may shed lights in further experimental investigation in this ?eld.

Acknowledgments

Agreement No. POD0410553.

This work is supported in part by an academic research fund of MOE, Singapore, and the DSTA under Project

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